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### LIST EDGE AND LIST TOTAL COLORINGS OF PLANAR GRAPHS WITHOUT 6-CYCLES WITH CHORD

Aijun Dong, Guizhen Liu, Guojun Li
2012 Bulletin of the Korean Mathematical Society
It is proved that if a planar graph G without 6-cycles with chord, then χ ′ l  ...  Giving a planar graph G, let χ ′ l (G) and χ ′′ l (G) denote the list edge chromatic number and list total chromatic number of G respectively.  ...  Planar graphs without 6-cycles with chord First let us introduce an important lemma. Proof. For G is a critical planar graph with ∆(G) ≥ 6, then G contains no (4, 4, 5 − )-face f = uvw.  ...

### 3-Choosability of Triangle-Free Planar Graphs with Constraints on 4-Cycles

Zdeněk Dvořák, Bernard Lidický, Riste Škrekovski
2010 SIAM Journal on Discrete Mathematics
A graph is k-choosable if it can be colored whenever every vertex has a list of at least k available colors. A theorem by Grötzsch  asserts that every triangle-free planar graph is 3-colorable.  ...  In addition, this implies that every triangle-free planar graph without 6-and 7-cycles is 3-choosable. * Supported by a CZ-SL bilateral project MEB 090805 and BI-CZ/08-09-005.  ...  This strengthens the results of Lidický  that planar graphs without 3-, 6-, 7-and 8-cycles are 3-choosable, and of Zhang and Xu  that planar graphs without 3-, 6-, 7-and 9-cycles are 3-choosable  ...

### Correspondence coloring and its application to list-coloring planar graphs without cycles of lengths 4 to 8 [article]

Zdenek Dvorak, Luke Postle
2016 arXiv   pre-print
Using this tool, we prove that excluding cycles of lengths 4 to 8 is sufficient to guarantee 3-choosability of a planar graph, thus answering a question of Borodin.  ...  We introduce a new variant of graph coloring called correspondence coloring which generalizes list coloring and allows for reductions previously only possible for ordinary coloring.  ...  On the other hand, planar graphs are 5-choosable  , and every planar graph without cycles of lengths 3 and 4 is 3-choosable  .  ...

### List Edge Colorings of Planar Graphs with 7-cycles Containing at Most Two Chords

Wenwen Zhang
2019 Applied Science and Innovative Research
In this paper we prove that if G is a planar graph, and each 7-cycle contains at most two chords, then G is edge-k-choosable, where k = max{8, ?(G) + 1}.  ...  Structural Properties of Planar Graphs with 7-Cycles Containing at Most Two Chords Lemma 4. Let G be a planar graph in which each 7-cycle contains at most two chords.  ...  Suppose that G is a counterexample to our theorem with the minimum number of edges and G is any planar graph in which every 7-cycle contains at most two chords.  ...

### On oriented cliques with respect to push operation [article]

Julien Bensmail, Soumen Nandi, Sagnik Sen
2015 arXiv   pre-print
To push a vertex v of a directed graph G is to change the orientations of all the arcs incident with v. An oriented graph is a directed graph without any cycle of length at most 2.  ...  We also provide an exhaustive list of minimal (with respect to spanning subgraph inclusion) planar push cliques.  ...  Note that an eight cycle can have three types of chords, namely, a very long chord connecting vertices at distance 4, a long chord and a short chord.  ...

### On Choosability with Separation of Planar Graphs with Forbidden Cycles

Ilkyoo Choi, Bernard Lidický, Derrick Stolee
2015 Journal of Graph Theory
We prove that planar graphs without 4-cycles are (3,1)-choosable and that planar graphs without 5-cycles and 6-cycles are (3,1)-choosable.  ...  We study choosability with separation which is a constrained version of list coloring of graphs.  ...  Jao for fruitful discussions and encouragement in the early stage of the project.  ...

### Algorithms [chapter]

2011 Graph Coloring Problems
Graphs 97 5.1 Critical Graphs With Many Edges 97 5.2 Minimum Degree of 4-and 5-Critical Graphs > 98 5.3 Critical Graphs With Few Edges 99 5.4 Four-Critical Amenable Graphs 101 5.5 Four-Critical  ...  of Hamilton Cycles . 82 4.6 Brooks' Theorem for Triangle-Free Graphs 83 4.7 Graphs Without Large Complete Subgraphs 85 4.8 ^-Chromatic Graphs of Maximum Degree k 85 4.9 Total Coloring 86  ...

### Do triangle-free planar graphs have exponentially many 3-colorings? [article]

Zdeněk Dvořák
2017 arXiv   pre-print
Thomassen conjectured that triangle-free planar graphs have an exponential number of 3-colorings.  ...  We show this conjecture to be equivalent to the following statement: there exists a positive real α such that whenever G is a planar graph and A is a subset of its edges whose deletion makes G triangle-free  ...  coloring of planar graphs.  ...

### Planar graphs without cycles of length 4, 7, 8, or 9 are 3-choosable

Yingqian Wang, Qian Wu, Liang Shen
2011 Discrete Applied Mathematics
It is known that planar graphs without cycles of length 4, i, j, or 9 with 4 < i < j < 9, except that i = 7 and j = 8, are 3-choosable.  ...  This paper proves that planar graphs without cycles of length 4, 7, 8, or 9 are also 3-choosable.  ...  Acknowledgements The authors are grateful to the referees for their wise suggestions to simplify the original proof of this paper.  ...

### Improper choosability of graphs of nonnegative characteristic

Yongzhu Chen, Weiyi Zhu, Weifan Wang
2008 Computers and Mathematics with Applications
In this paper, we prove: (1) If G contains no k-cycle with a chord for all k = 4, 5, 6, then G is (3, 1) *choosable; (2) If G contains neither 5-cycle with a chord nor 6-cycle with a chord, then G is (  ...  A graph G is called (k, d) * -choosable if, for every list assignment L with |L(v)| = k for all v ∈ V (G), there is an L-coloring of G such that every vertex has at most d neighbors having the same color  ...  Equivalently, every torodial graph without a 4-cycle with a chord is (4, 1) * -choosable. Theorems 5 and 6 are, to some extent, an extension to their result.  ...

### List coloring of planar graphs without 6-cycles with two chords

Linna Hu, Lei Sun, Jian-Liang Wu
2018 Discussiones Mathematicae Graph Theory
If ∆(G) ≥ 6, then χ ′ list (G) ≤ ∆(G) + 1. The following result is about edge-∆-choosable of embedded planar graphs without 6-cycles with two chords. Theorem 8.  ...  Main Results and Their Proofs In the section, we always assume that all graphs are planar graphs that have been embedded in the plane and G is a planar graph without 6-cycles with two chords.  ...  By Lemmas 9 and 10, v may be the 3master of two 3 − -vertices and the 2-master of a 2-vertex, that is, v sends at most 5 to its incident 3 − -vertices. Suppose that d(v) = 9. Then f 3 (v) ≤ 6.  ...

### (4, 2)-Choosability of Planar Graphs with Forbidden Structures

Zhanar Berikkyzy, Christopher Cox, Michael Dairyko, Kirsten Hogenson, Mohit Kumbhat, Bernard Lidický, Kacy Messerschmidt, Kevin Moss, Kathleen Nowak, Kevin F. Palmowski, Derrick Stolee
2017 Graphs and Combinatorics
A chorded ℓ-cycle is an ℓ-cycle with one additional edge. We demonstrate for each ℓ∈{5,6,7} that a planar graph is (4,2)-choosable if it does not contain chorded ℓ-cycles.  ...  A graph is (k,s)-choosable if the graph is colorable from lists of size k where adjacent vertices have at most s common colors in their lists.  ...  Conclusion and Future Work We proved that, for each k ∈ {5, 6, 7}, planar graphs with no chorded k-cycles are (4, 2)-choosable.  ...

### On oriented cliques with respect to push operation

Julien Bensmail, Soumen Nandi, Sagnik Sen
2017 Discrete Applied Mathematics
An oriented graph is a directed graph without any directed cycle of length at most 2. An oriented clique is an oriented graph whose non-adjacent vertices are connected by a directed 2-path.  ...  We also prove that a planar push clique can have at most 8 vertices and provide an exhaustive list of planar push cliques.  ...  Acknowledgement: The authors would like to thank the anonymous reviewer for the constructive comments towards improvement of the content, clarity and conciseness of the manuscript.  ...

### Planar graphs without normally adjacent short cycles [article]

Fangyao Lu and Mengjiao Rao and Qianqian Wang and Tao Wang
2021 arXiv   pre-print
Consequently, every planar graph without 4-, 6-, 8-cycles is 3-choosable, and every planar graph without 4-, 5-, 7-, 8-cycles is 3-choosable.  ...  Let 𝒢 be the class of plane graphs without triangles normally adjacent to 8^--cycles, without 4-cycles normally adjacent to 6^--cycles, and without normally adjacent 5-cycles.  ...  This work was supported by the National Natural Science Foundation of China and partially supported by the Fundamental Research Funds for Universities in Henan (YQPY20140051).  ...